New J. Phys. 21 ( 2019 ) 103025 https: // doi.org / 10.1088 / 1367-2630 / ab42ae
PAPER
Photon-number parity of heralded single photons from a Bragg-
re fl ection waveguide reconstructed loss-tolerantly via moment
generating function
K Laiho
1 , 6
, M Schmidt
1 , 2
, H Suchomel
3
, M Kamp
3
,SH ö fl ing
3 , 4
, C Schneider
3
, J Beyer
2
, G Weihs
5
and
S Reitzenstein
1
1
Technische Universität Berlin, Institut für Festkörperphysik, Hardenbergstr.  36, D-10623, Berlin, Germany
2
Physikalisch-Technische Bundesanstalt, Abbestr.  2-12, D-10587, Berlin, Germany
3
Technische Physik, Universität Würzburg, Am Hubland, D-97074, Würzburg, Germany
4
School of Physics & Astronomy, University of St Andrews, St Andrews, KY16 9SS, United Kingdom
5
Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020, Innsbruck, Austria
6
Author to whom any correspondence should be addressed.
E-mail: [email protected]
Keywords: factorial moment of photon number, photon-number parity, moment generating function, parametric down-conversion,
Bragg-re fl ection waveguide, transition-edge sensor
Abstract
Due to their strict photo n-number correlation, the t win beams produced in parametric down-conversion
( PDC ) work well for he ralded state generation. Often, however, this state m anipulation is dis torted by the
optical losses in the herald and by t he higher photon- number contributions inevitable in the PDC process.
In order to fi nd feasible fi gures of merit for characterizing the heralded s tates, we investigate their normalized
factorial moments of the photon number that can be acce ssed regardless of the optical losses in the detection.
We then perform a measurement of the j oint photon statistics of twin beams from a sem iconductor Bragg-
re fl ection waveguide with transition-edge sensors actin g as photon-number-resolving detectors. We extract
the photon-number parity of heralded single photons in a loss-tolerant fashion by utilizing the moment
generating function. The photon-number parity is high ly practicable in quantum state characterization,
since it takes into account the complete photon-number content of the target state.
1. Introduction
The process of parametric down-conversion ( PDC ) , which generates pairs of photons, has proven to be
expedient for heralding single photons [ 1 , 2 ] as well as higher photon-number states [ 3 – 5 ] . When pumped with a
bright laser, the twin beams from the PDC process inevitably include higher photon-number contributions, the
characteristics of which have been investigated in the past via the normalized Glauber correlation functions
[ 6 – 8 ] . Luckily, these correlation functions also provide valuable information of the PDC process parameters [ 9 ] .
Alternatively, the higher-order moments can be extracted from the joint photon-number distribution of the
twin beams [ 10 , 11 ] . Moreover, such direct measurements of their joint photon statistics also deliver access to
the non-classicality of different classes of heralded states [ 12 , 13 ] .
Due to the experimental imperfections, like the optical losses both in the state manipulation and target state
detection, the measured photon statistics are often degraded and the fragile quantum features are lost [ 14 , 15 ] .I n
order to decouple the effect of these two loss contributions one can investigate the target state properties with the
help of the normalized factorial moments of the photon number that can be extracted independent of the
detection losses. Therefore, higher-order moments are routinely used for loss-independent veri fi cation of the
quantum characteristics of radiation fi elds [ 16 – 19 ] . Additionally, when combined with the loss-tolerant
determination of the mean photon number of the studied target state, this method can be employed for a more
direct quantum state characterization via the so-called moment generating function [ 20 ] , which provides, for
example, access to the individual photon-number contributions and even to the photon-number parity [ 21 – 24 ] .
OPEN ACCESS
RECEI VED
28 June 2019
REVISED
28 August 2019
ACCEPTED FOR PUBLICATION
9 September 2019
PUBLISHED
9 October 2019
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© 2019 The Author ( s ) . Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft

Until today, only few ( non-commercial ) photodetectors provide true photon-number resolution at the
single- and few photon level. Such detectors are, for example, the visible light photon counter [ 4 ] and the
superconducting transition-edge sensor ( TES ) [ 25 , 26 ] . The latter is a very sensitive calorimeter working at the
transition edge between the superconducting phase and the normal phase. Apart from being practicable in
photon counting, that is, reconstructing the photon statistics of the measured light [ 27 – 29 ] , the TES-based
detectors have also been employed in spectral measurements due to their excellent energy resolution [ 30 ] .
Besides, a quantum optical tomography of the TES has veri fi ed its ability to work as a linear photon counter
without spurious noise contributions [ 31 ] .
Here, we employ a Bragg-re fl ection waveguide ( BRW ) fabricated of AlGaAs to generate broadband type-II
PDC emission with a moderate photon-pair correlation [ 32 , 33 ] for producing heralded single photons [ 34 ] .
Importantly, we use TES-based photon-number-resolving detectors for measuring the emitted joint photon
statistics of twin beams. To easily visualize the characteristics of the heralded states we fi rst theoretically
investigate their higher-order moments in terms of the optical losses in the herald and the mean photon number
of the PDC emission [ 35 , 36 ] , both of which are easily accessible in a measurement. Thereafter, we explore the
loss-tolerant reconstruction of the photon-number parity via the moment generating function providing a
boundary for the reliable state reconstruction region in a real experiment. Altogether, our investigations give
insight in the suitability of the twin-beam based photon-pair sources to heralding tasks. Further, we provide a
direct connection between the heralded state quality and the important parameters of the state generation and
manipulation. We believe that our straightforward method for the loss-tolerant reconstruction of the photon-
number parity can become practical in quantum state characterization.
2. Theory
We start by investing heralding as sketched in fi gure 1 with a well-behaved PDC process, which generates
perfectly photon-number correlated twin beams called signal and idler. This state can be expressed as
nn ,, 1
n
ns i
0
å
yl ñ= ñ
=
¥
∣∣ ( )
in which n denotes the photon number in signal ( s ) and idler ( i ) and
n

2
l
∣

∣ is the weight of each photon-number
contribution. Given PDC with a mean photon number of n
á

ñ ˜ , the weight distribution in equation ( 1 ) is thermal
i.e.
nn 1
n n
n

2 1
l =á ñ +á ñ
+
∣

∣˜ ( ˜ )
, if it emits into a single mode ( SM ) [ 37 ] , and turns into Poissonian i.e.
nn n exp
n n
2
l = - áñ áñ
∣

∣( ˜ ) ˜ !
, if the PDC emission is highly multimodal ( MM ) [ 38 ] .
In order to extract the characteristics of the heralded target state ( here signal ) , we trace ( Tr ) the density
matrix
si


yy =ñ á ∣∣
over the herald ( here idler ) as
Tr
Tr ,2
is i si
si s i si


 

= ÄP
ÄP
s
s
{( ˆˆ
)}
{( ˆˆ
)} ()
in which the operators i
P s

ˆ describe the photodetector used in heralding ( σ labeling the different measurement
outcomes ) and form a complete set i

å

P=
s
s
ˆ
. The photon statistics of the heralded target state is then given as
a projection to the photon-number basis spanned by the photon-number states n ñ
∣

and the probability of the n th
photon-number contribution can be expressed as
pn n n Tr
ss s

= () { ∣ ⟩ ⟨ ∣ }
.
A loss-degraded photon-number-resolving detector ( σ  =  n ) used in heralding can be described by
N
n NN 1, 3
i
n
Nn
Nn n ii

å
hh P= - ñ á
-
()
ˆ () ∣ ∣ ( )
in which η denotes the detector ef fi ciency. For comparison we also examine the loss-degraded bucket detector at
heralding ( σ  =  click or no click ) , whose properties are ideally given by
Figure 1. Parameters involved in heralding with well-behaved PDC. The mean photon number of the PDC emission n
áñ

˜ is
proportional to the optical pump power P . The twin beam, which is used as herald, is detected with a lossy photon counter that is
modeled by placing a beam splitter with the transmittance η corresponding to the heralding ef fi ciency in front of an ideal detector that
projects the state onto the photon-number basis nn ñá
∣

∣ . The heralded target state ñ with a mean photon number of n
áñ

is prepared in
the other twin beam whenever a herald is detected.
2
New J. Phys. 21 ( 2019 ) 103025 K Laiho et al

NN
1a n d
1. 4
i
click
i
no click
i
no click
N
N ii
^^
^
å
h
P= - P
P= - () ∣ ⟩ ⟨ ∣ ()
For heralding single photons we can either choose σ  =  1 ( equation ( 3 )) or σ  =  click ( equation ( 4 )) . These
are plugged into equation ( 2 ) and we calculate the resulting photon statistics. After that we employ the
normalized factorial moments of the photon number for the state characterization. These moments can be
extracted from the photon statistics via [ 20 , 21 ]
g n
n
nn n m pn
np n
:: 1 ... 1 ,5
m m
m
n
n
m
å
å
= áñ
áñ = -- +
()
ˆ
ˆ
() ( ) ( )
()
()
()
with
n

nn n
n
= å ñá
ˆ ∣∣
being the operator delivering the mean photon number and :: denoting the normal-
ordering of operators. In fi gure 2 we illustrate the g
( 2 )
-function of the heralded target states in both investigated
cases in terms of the heralding ef fi ciency and the mean photon number of the PDC emission. The differences in
the target state characteristics when heralded either with a true photon-number-resolving detector in fi gure 2 ( a )
or with a bucket detector in fi gure 2 ( b ) are clearly visible. It is still possible to achieve g
( 2 )
 ≈  0, with either
heralding methods regardless of the heralding ef fi ciency if the mean photon number of the PDC emission, i.e.
the pump power is kept low. Additionally, the results for SM and MM PDC emission provide boundaries for the
achievable g
( 2 )
-values with well-behaved real PDC sources. Most interestingly, however, we see that a true
photon-number-resolving detector provides advantages over the bucket detector and can clearly help
suppressing the effect of the higher photon-number contributions if a high heralding ef fi ciency exceeding about
0.7 is achieved, which helps keeping g
( 2 )
   0.5 until
n 1
á

ñ» ˜
for both SM and MM PDC. Further, we note that
when regarding the generation of higher photon-number states with true photon-number-resolving detectors
the situation remains very similar as for heralding single photons: thus, either a very low pump power or a high
heralding ef fi ciency is required in order to be able to generate the desired photon-number content in the
heralded target state.
The higher-order normalized factorial moments provide an interesting insight into the state characteristics,
and they are connected to the so-called moment generating function [ 20 , 21 ]
Mp n
g
m n 16
n
n
m
m m
åå
mm m =- = - á ñ () ( ) ( ) ! () ( )
()
given in terms of a real-valued variable 0    μ    2 related to the operator ordering. Here, we are merely
interested in reconstructing the quantum state characteristics via the right-hand side of equation ( 6 ) . For that
purpose, apart from knowing the higher-order normalized moments also the mean photon number of the target
state
n
á

ñ
needs to be measured loss-tolerantly. In general, the mean photon number can be corrected simply by
dividing the measured, loss-degraded mean photon number of the target state with the ef fi ciency, with which it
is being detected [ 20 , 23 ] . Looking further into equation ( 6 ) , the value of the moment generating function at
μ  =  2 delivers the photon-number parity , for which − 1    M ( 2 )    1. Being the difference between the weights
of the even and odd photon-number contributions, the photon-number parity is an important fi gure of merit
in quantum optics [ 39 ] . A negative photon-number parity — ideally taking the value of M ( 2 )  =  − 1 for single
photons — offers a more stringent criterium of the state ’ s non-classicality than the higher order moments
Figure 2. The g
( 2 )
-function of the heralded target state calculated via equation ( 5 ) , when conditioned with a measurement of ( a ) one
photon with a photon-number-resolving detector and ( b ) a single click with a bucket detector in the herald. Since the latter can only
resolve between click and no click , it heralds in all cases in which one or more photons impinge on it. Orange and blue surfaces as
labeled in the fi gure illustrate the behavior of SM and MM PDC emission, respectively.
3
New J. Phys. 21 ( 2019 ) 103025 K Laiho et al

do [ 40 ] . Especially, the non-classicality criterium g
( 2 )
 <  1, which can be related to the width of the photon-
number distribution being sub-Poissonian, is commonly employed only to indicate to which extent photon-
number contributions higher than or equal to two are existent.
We note that extracting the photon-number properties via the alternating sum on the right-hand side of
equation ( 6 ) may not necessarily converge, if the values of the normalized factorial moments grow strongly with
increasing order of m , like for thermal states. However, when regarding certain quantum optical states such as
single photons it can be practical. In fi gure 3 we investigate the reconstruction of the photon-number parity of
single-photon states heralded from MM PDC. We fi rst calculate the target state characteristics at a low heralding
ef fi ciency comparable to the one in our experiment. By combining the mean photon number of the heralded
state in fi gure 3 ( a ) with the normalized factorial moments in fi gure 3 ( b ) , the photon-number parity can be
reconstructed. As illustrated in fi gure 3 ( c ) the useful reconstruction range for the photon-number parity
strongly depends on the order of the highest recorded moment. Second, we perform a calculation of the photon-
number parity also at higher heralding ef fi ciencies in fi gure 3 ( d ) , which highlights its usefulness as a fi gure of
merit of the heralded state quality and, when mapped in terms of the parameter space ( η , n
á

ñ ˜ ) , it denotes the
regions, in which high quality single photons can be created. In contrast to the often employed photon-number
fi delity, which only includes the targeted photon-number contribution, the photon-number parity also includes
information of the undesired higher photon-number contributions. Finally, fi gure 3 ( e ) illustrates the target state
preparation probability determined as
T

r11
si s s i
n
si
1
1 2

hl ñá Ä P =
=
{(∣ ∣ ˆ )} ∣ ∣
, which denotes the probability of
the successful heralding with the desired outcome in the target state. Although we found out in fi gure 3 ( d ) that
high quality target states can also be prepared at low values of n
á

ñ ˜ , fi gure 3 ( e ) shows that the state preparation in
this region is very rare. As expected, the region of enlarged yield lies at high heralding ef fi ciencies close
to
n 1
á

ñ= ˜
.
3. Experiment
The investigated ridge BRW has a width o f
4

.5 m m and a leng th of
1

.7 mm . O ur BRW incorpora tes the structure
fi rst presented in [ 41 ] and supports a collinear type-II PDC process. Thence, the pump photons in the Bragg-mode,
Figure 3. Theoretical investigation of the photon-number characteristics of single photons heralded from MM PDC. ( a ) The mean
photon number, ( b ) the normalized higher-order moments up to the 5th order and ( c ) the photon-number parity reconstructed via
the right-hand side of equation ( 6 ) when truncated to the m th order are presented in terms of the mean photon number of the PDC
emission at a heralding ef fi ciency on the order of few percent. The solid black line in ( c ) shows the expected photon-number parity
calculated via the left-hand side of equation ( 6 ) .I n ( d ) we illustrate the photon-number parity and in ( e ) the target state preparation
probability in terms of the heralding ef fi ciency and mean photon number of PDC emission. While ( d ) highlights the necessity to
control both these parameters in order to approach the desired value of − 1, our results in ( e ) show the regions, where heralding is
most ef fi cient.
4
New J. Phys. 21 ( 2019 ) 103025 K Laiho et al

which is a hi gher-order spatial mode, split in to cross-pol arized signal an d idler phot ons both i n total-internal
re fl ection modes. Our BRW samples are gr own w ith molecular beam epitaxy and th e rid ge s ar e p at t er n ed b y h i gh -
resolution el ectron beam lithography follow ed by react ive-ion plasma etching. The BRWs are etched j ust above the
core and fi nally pass ivated with a polymer on top. T he cleaved wavegu ide facets ar e kept uncoated.
In our experiment depicted in fi gure 4 ( a ) we use approximately
1

.2 ps
long Ti:Sapphire laser pulses with a
central wavelength of
7

71 nm
and about
0

.5 nm full-width-at-half-maximum as the pump for the PDC process.
The pump beam is sent through an acusto-optic modulator ( AOM ) in order to reduce its repetition rate to
7

8k H z and the pump power is controlled with a variable neutral density fi lter ( NDF ) . Thereafter, the pump
beam is sent through a polarization-maintaining single-mode fi ber to the BRW setup. We use a half-wave plate
( HWP ) and a removable sheet polarizer ( not shown ) for selecting the proper pump beam polarization. After
passing through a short-pass fi lter ( SPF ) , the pump beam is coupled to the BRW for producing PDC with an
aspheric lens ( AL ) having a focal length of approximately
3

mm
.
Another AL collimates the PDC emission from our BRW, after which the pump beam is separated from the
beam path with a dichroic mirror ( DM ) . Thereafter, the PDC emission is sent through a long-pass fi lter ( LPF ) as
well as a
1

2n m
broad bandpass fi lter ( BPF ) centered close to the degeneracy wavelength of
1

542 nm , which we
previously measured via the second-harmonic generation. Signal and idler then pass through another HWP and
get separated in a polarizing beam splitter ( PBS ) . Finally, we couple the individual twin beams into single-mode
fi bers with ALs having about
1

6m m fi xed focus, with which we achieve  50% coupling ef fi ciency near
1

550 nm . After that signal and idler are recorded with two TESs optimized for photodetection in the
telecommunication wavelength range [ 25 ] .
The two-channel photon-number-resolving detector system includes two TESs as the light-sensitive
detector elements. Each TES consists of a thin fi lm of tungsten embedded in a multilayer structure, in which the
individual photons are absorbed [ 42 ] . Our TESs are kept in a demagnetization refrigerator close to
1

00 mK
with
a holding time of about 25 h [ 26 ] . The individual TESs reach detection ef fi ciencies above 60% and 70%, which
were measured with a modulated continuous-wave laser in the telecommunication wavelengths. Finally, the
TESs ’ responses are ampli fi ed with superconducting quantum interference devices [ 43 ] before the detected
Figure 4. ( a ) Experimental setup for measuring the joint photon-number distribution of twin beams from a BRW with TESs,
( b ) graphic presentation of 30 000 raw traces collected in about
0

.38 s
and measured at a pump power of
1

7.6 W m
by an individual
TES, ( c ) histogram of the area of the raw TES traces measured in
1

0.4 s
at the pump power of
1

7.6 W m
, ( d ) – ( e ) counting rates of the
two TESs with respect to time elapsed from the trigger at different pumping powers gained by post-processing the TES traces like in ( b )
by discriminating them with a level just above the vacuum base line and ( f ) the complete measured joint photon-number distribution
at
1

7.6 W m
. The horizontal ticks in ( c ) between the minimum and maximum trace area, A
min.
and A
max.
respectively, calculated from
the traces including those in ( b ) indicate the separation between the vacuum, one- and two-photon contributions. The dotted lines in
( d ) – ( e ) illustrate the boundaries of the used time gates. The signal-to-noise ratio in ( d ) – ( e ) is on the order of 20 for both TESs being the
ratio between the maximal count rate near
1

s m
and the count rate near
5

.5 s m
. The gray areas in ( f ) illustrate the measurement
uncertainties in the joint photon statistics ( blue bars ) . Abbreviations: AL  =  aspheric lense; AOM  =  acusto-optic modulator;
BPF  =  bandpass fi lter; BRW  =  Bragg-re fl ection waveguide; DM  =  dichroic mirror; HWP  =  half-wave plate; LPF  =  long-pass
fi lter; NDF  =  neutral density fi lter; PBS  =  polarizing beam splitter; SPF  =  short-pass fi lter.
5
New J. Phys. 21 ( 2019 ) 103025 K Laiho et al

traces are digitized with an analog-to-digital converter as shown in fi gure 4 ( b ) . A histogram of the raw events
detected by an individual TES in fi gure 4 ( c ) clearly shows vacuum, one- and two-photon contributions in a twin
beam. Further, we utilize time gates of about
1

.5 s m as illustrated in fi gures 4 ( d ) – ( e ) to reduce background events
from the rather strong photoluminescence coming from the BRW wafer material ( see fi gure 4 ( b )) . The electrical
trigger from the AOM is delivered for the synchronization of the measured traces. In order to acquire the joint
photon statistics as illustrated in fi gure 4 ( f ) we record ( 6.6 − 9.8 ) × 10
6
traces corresponding to effective
measurement times from 84 to
1

25 s and resulting up to about 5 GB of raw binary data.
4. Results
Due to the strong spectral correlation between signal and idler our BRW produces by de fi nition MM PDC
emission [ 33 , 44 ] . As we are only interested in the photon-number degree of freedom, our BRW is suitable for
the investigations, and we measure the joint photon statistics of the PDC emission by varying the pump power.
To prove that photon-pair generation takes place, we fi rst investigate the signal-idler correlation with the
coincidences-to-accidentals ratio ( CAR ) , which can be extracted loss-independently from the measured loss-
degraded joint photon statistics ( see appendix A ) . Our results in fi gure 5 ( a ) show, as expected, a decreasing
photon-pair correlation with increasing pump power. Indeed, a value CAR  >  2 is required to verify that the
correlation between signal and idler is stronger than that of a thermal light source. We achieve a maximum of
CAR  =  8.2 ( 3 ) that is well above this limit. Moreover, the photon-pair correlation provides a direct loss-
independent access to the mean photon number of the PDC emission, which can be estimated in the
MM case via n CAR 1 1 =+ á ñ ˜ [ 33 , 45 ] . This is shown for our data in the inset in fi gure 5 ( a ) delivering
n 0.139 5 ... 2.13 9
á

ñ= ˜( ) ( )
for the range of the measured pump powers and providing in our case the lower limit
for the mean photon number of the PDC emission.
We then investigate the characteristics of the higher-photon number contributions of the heralded target
states gained from the joint photon statistics when conditioned on measuring one photon with the TES in the
heralding arm. In fi gure 5 ( b ) we illustrate the evaluated g
( 2 )
-values of the heralded states both in signal and idler
in terms of the extracted mean photon number of the PDC emission. As expected, at low heralding ef fi ciencies
( see inset in fi gure 5 ( b )) , the extracted values of g
( 2 )
increase quickly with respect to the growing mean photon
number of the PDC emission. Moreover, the measured values agree well with the theoretical prediction in the
MM case. In our case the preparation of high quality heralded states is strongly limited to the low values of n
á

ñ ˜ ,
i.e. to the low pump powers, due to the modest heralding ef fi ciency achieved. To improve the heralding
ef fi ciency a better suppression of spurious counts via more stringent time-gating [ 46 ] , a more ef fi cient coupling
of the total-internal re fl ection modes into the single-mode fi bers, and a reduction of the BRW ’ s internal losses is
required [ 48 ] . Nevertheless, the preparation of single photons with close to the desired photon-number content
is still possible in our case, if the mean photon number of the PDC emission is kept low.
Therefore, in order to extract the photon-number parity of the heralded states via the moment generating
function, we perform a second set of power dependent measurements of the joint photon statistics at low pump
powers. Again, we evaluate the signal-idler correlation as shown in fi gure 6 ( a ) . At the lowest pump power we
Figure 5. ( a ) Measured CAR ( gray circles ) with respect to the power of the pump beam measured right before being coupled to the
BRW. If not shown, error bars are smaller than used symbols. The dashed line is a fi t, for which P CAR
1.19 10
µ
- ()
. An exponent of − 1
is expected for a well-behaved MM PDC process without spurious counts [ 46 ] . The dotted line highlights the value of unity expected
for two completely independent photon fl uxes. The inset in ( a ) shows the extracted mean photon number of the PDC emission
versus  pump power. ( b ) Evaluated g
( 2 )
-values of the heralded states in terms of the mean photon number of the PDC emission for
both signal ( blue diamonds ) and idler ( red squares ) , when the other twin beam is used as herald. The solid line is the theoretical
prediction for g
( 2 )
from fi gure 3 ( b ) . The inset in ( b ) shows the Klyshko ef fi ciencies of signal and idler [ 47 ] , which are evaluated as the
ratio of coincidence counts to singles counts in the other twin beam, and are used as an estimate for the heralding ef fi ciency.
6
New J. Phys. 21 ( 2019 ) 103025 K Laiho et al

achieve a CAR of 12.6 ( 2 ) limiting the mean photon number of the PDC emission to
n 0.086 2
á

ñ= ˜( )
.I n
fi gure 6 ( b ) we illustrate the extracted g
( 2 )
-values for the heralded target states with respect the pump power and
mark their loss-corrected mean photon numbers ( see appendix B ) . At the lowest pump power we reach the
values of g
( 2 )
 =  0.21 ( 3 ) and
n 1.086 2
á

ñ= ()
. Indeed, both of these fi gures of merit re fl ect the existing higher
photon-number content in the heralded state.
Due to the low yield of heralded state preparation and the limited acquisition time in our experiment, we
have at the lower pump powers experimentally access only to the second-order normalized factorial moment of
the heralded states. In order to loss-tolerantly evaluate their photon-number parity, we truncate the summation
in equation ( 6 ) to m  =  2. This delivers
M

nn g 21 2 2
,

22
»- á ñ + á ñ () () to which the data from fi gure 6 ( b ) is
plugged and the extracted values for M ( 2 ) are shown in fi gure 6 ( c ) with respect to the mean photon number of
the PDC emission. Even if only the second-order normalized moment has been measured, a negative photon-
number parity can be reconstructed up to
n 0.
4


á

ñ ˜
for well-behaved MM-PDC at low heralding ef fi ciencies as
shown by the dashed – dotted line in fi gure 6 ( c ) and our results nicely follow this tendency. At the lowest pump
power we can reconstruct the photon-number parity of − 0.68  ±  0.08 for the heralded state. However, one
clearly sees that the truncation of equation ( 6 ) , which is necessary due to the experimental limitations, causes
distortion from the theoretical expectation.
5. Conclusions
We measured the joint photon statistics of twin beams generated via parametric down-conversion in BRWs with
TESs, which are true photon-number-resolving detectors. We showed that the normalized factorial moments of
the photon number are expedient for providing an easy access to the heralded state characteristics in terms of the
parameter space spanned by the mean photon number of the PDC emission and the heralding ef fi ciency. By
combining the measured second-order normalized moment of the heralded single photon with its mean photon
number corrected for experimental imperfections, we were able to reconstruct its negative photon-number
parity. We believe our experimental method can become powerful in order to loss-tolerantly investigate the
characteristics of also more sophisticated states, like higher photon-number states. Additionally, our scheme is
not limited to true photon-number-resolving detectors but can also be implemented with bucket detectors
connected to cascaded optical beam splitter networks, which are also capable of measuring higher-order
moments.
Acknowledgments
We acknowledge support from the Austrian Science Fund ( FWF ) : J-4125-N27 and I-2065-N27, the German
Research Foundation ( DFG ) : RE2974 / 18-1 and SCHN1376 / 2-1 and the State of Bavaria. The work reported in
this paper was partially funded by project EMPIR 17FUN06 SIQUST. This project has received funding from the
EMPIR programme co fi nanced by the Participating States and from the European Union ʼ s Horizon 2020
research and innovation program.
Figure 6. Experimental values ( symbols )( a ) for the CAR and ( b ) for g
( 2 )
of the heralded target states with respect to the pump power as
well as ( c ) for the loss-tolerantly reconstructed photon-number parity with respect to the mean photon number of the PDC emission.
The dashed line in ( a ) is fi tted with P CAR
1.34 11
µ
- ()
and the dotted line shows CAR  =  1. The dashed line in ( b ) is a linear fi t and the
loss-corrected mean photon numbers n
áñ

of the heralded target states marked are extracted from the values in ( a ) . The reconstructed
values in ( c ) agree well with equation ( 6 ) when truncated to m  =  2 ( dashed – dotted line ) . In order to reconstruct the expected photon-
number parity from fi gure 3 ( c )( black solid line ) , a measurement of normalized moments with orders much higher than two would be
required. If not shown, error bars are smaller than used symbols.
7
New J. Phys. 21 ( 2019 ) 103025 K Laiho et al

We gratefully thank F Gericke for laboratory support, A Wolf and S Kuhn for assistance during the sample
growth and fabrication, M López from PTB Braunschweig for lending us a tunable telecom continuous-wave
laser, and A E Lita and S W Nam from NIST, USA, for providing us the transition-edge sensor chips.
Appendix A. Extracting signal-idler photon-number correlation from the joint photon
statistics
The higher-order normalized moments can be extracted directly from the measured joint photon statistics of
twin beams. We calculate them directly via [ 7 ]
g nn
nn
::
,A . 1
mn s
m i n
s m i n
,
= áñ
áñ á ñ
ˆˆ
ˆˆ ()
()
in which ( m , n ) denotes the order of the correlation and
n

s
ˆ
and
n i

ˆ
are the mean photon-number operators in the
signal and idler arms, respectively. We evaluate the moments in the photon-number basis, and therefore assume
that the density matrix of the two-mode state can be written as ck k l l
si kk ll kk ll ss ii
,,
,, ,


= åå ñá
¢ Äñ á
¢
¢¢ ¢¢
∣∣ ∣
∣

. The
joint photon statistics can then be expressed as
P

kl k k l l c c ,T r
s i ss ii kkl l kl ,, , ,

=ñ á Ä ñ á = = () { ∣ ∣ ∣ ∣ } ˜
. Due to
the normalization
Pk l c ,1
kl kl
, ,
å

== () ˜
.
We can evaluate the higher-order moments by re-writing
n

aa
s
=
ˆˆ ˆ
†
and
n

bb
i
=
ˆ ˆ
ˆ

†
with
a

ˆ (
a

ˆ † ) and
b
ˆ
( b
ˆ † )
being the photon annihilation and creation operators in signal and idler arms. The expectation values involved
in equation ( A.1 ) can the be evaluated with the help of
nn a b b a
ck a a k l b b l
ck k k m l l l n
kc l
:: T r
1 ... 1 1 ... 1
,A . 2
s
m i n si m nn m
kk ll
kk ll s mm si
nn
i
kl
kl
kl
kl
kl
,,
,, ,
,
,



åå
åå
åå
áñ =
=á
¢ ñá
¢ ñ
=- - + - - +
=
¢¢
¢¢
==
         
ˆˆ { ˆ ˆˆ
ˆ }
∣ ˆˆ
∣∣
ˆˆ
∣
˜ ( )( ) ( )( )
() ˜ ( ) ( )
†
† †
() ( )
the fi nal form of which can be evaluated as a matrix multiplication. In section 4 we utilize the signal-idler
correlation g
( 1,1 )
, which corresponds to the well-known CAR.
Appendix B. Accessing the mean photon number of heralded state
The signal-idler c orrelations can be very he lpful in chara cterizing the properties of heralded t arget states and in deed,
they are useful for estimating the mea n photon number of the heralded states in a loss-tolerant manner. Because of the
low detection ef fi cie ncy, we can make use of coincidence counting for approximati ng it. Thus, the ratio of coincidence
counts C to single coun ts S
ξ
( ξ  =  s , i ) for signal ( s ) and idler ( i ) provides a good approximation for the loss-degraded
mean photon number of the heralded state, which is given by
nC S
si is
,
lossy ,
á

ñ»
. The loss-invert ed mean photon
number can be extracted as nn
si si s
i

, ,
lossy
,
h
á

ñ= á ñ ¢ with
CA S
si is
, ,
h

¢ =- ()
being the effective Kl yshko ef fi ciency of
signal and idle r, which accounts for the eff ect of higher ph oton-number contributions since the accidental counts A
are subtracted fr om the measured coincidenc es. Therefore , t he loss-tolerantly determine d mean photon number of
the heralded sta te can be estimated with the help of the CAR denoted as C / A via n 1
si
C
S
S
CA C A
,
1 1
is
is
,
,
á

ñ» = -
-
-
()
.
ORCID iDs
K Laiho https: / / orcid.org / 0000-0003-3090-8629
SH ö fl ing https: / / orcid.org / 0000-0003-0034-4682
G Weihs https: / / orcid.org / 0000-0003-2260-3008
S Reitzenstein https: / / orcid.org / 0000-0002-1381-9838
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